Semidefinite games

TL;DR

This paper introduces semidefinite games, a generalization of classical game models using positive semidefinite matrices, and explores their properties, including computation of equilibria and the structure of Nash equilibria.

Contribution

It extends game theory to semidefinite strategies, providing computational methods and characterizations for Nash equilibria in this new framework.

Findings

Optimal strategies in semidefinite zero-sum games can be computed via semidefinite programming.

Two-player semidefinite zero-sum games are nearly equivalent to semidefinite programming.

Constructed semidefinite games with more Nash equilibrium components than previous models.

Abstract

We introduce and study the class of semidefinite games, which generalizes bimatrix games and finite $N$-person games, by replacing the simplex of the mixed strategies for each player by a slice of the positive semidefinite cone in the space of real symmetric matrices. For semidefinite two-player zero-sum games, we show that the optimal strategies can be computed by semidefinite programming. Furthermore, we show that two-player semidefinite zero-sum games are almost equivalent to semidefinite programming, generalizing Dantzig's result on the almost equivalence of bimatrix games and linear programming. For general two-player semidefinite games, we prove a spectrahedral characterization of the Nash equilibria. Moreover, we give constructions of semidefinite games with many Nash equilibria. In particular, we give a construction of semidefinite games whose number of connected components of Nash equilibria exceeds the long standing best known construction for many Nash equilibria in bimatrix games, which was presented by von Stengel in 1999.